Despite adding centre-backs Connor Goldson and Nikola Katic, Rangers are still keen to sign another.
The Gers have made a £3 million offer to land Millwall's Jake Cooper, SkySports reported.
If the funds are available, just £1 million more would allow Rangers a chance to land former Ibrox loan star Kyle Bartley.
Bartley is close to leaving Swansea City for Championship rivals West Bromwich Albion.
SkySports report an agreement between the clubs is close over a £4 million transfer.
Rangers were said to be keen on Bartley earlier this summer and the Scottish Sun claimed he was open to a return to Ibrox.
It was during his time as an Arsenal trainee in 2011/12 that Bartley spent time on loan at Rangers.
He impressed, and memories of him scything down Scott Brown in an Old Firm game live on for fans.
West Brom are in pole position to land the defender, but for just a little bit more than they are currently willing to pay for Cooper, Rangers should make a last push to try and land Bartley instead.